reconstruction synthesis of the lost subsystem for the planetary motions of antikythera...
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Taipei Citys
Taipei City
Tainan City
Creative Machine Design E&R Lab, National Cheng Kung University, TAIWAN
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ASME 2011 International Design Engineering Technical Conferences (IDETC) and Computers and Information in
R S h f h L
Conferences (IDETC) and Computers and Information in Engineering Conference (CIE)
Reconstruction Synthesis of the Lost Subsystem for the Planetary Motions Subsystem for the Planetary Motions
of Antikythera Mechanism
Authors: Hong-Sen Yan & Jian-Liang LinSpeaker: Jian-Liang Lin
Department of Mechanical EngineeringNational Cheng Kung University
Creative Machine Design E&R Lab, National Cheng Kung University, TAIWAN
Tainan, Taiwan
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Outline
IntroductionOverall ContributionHistorical BackgroundHistorical BackgroundKinematic Analysis i i SFeasible Topological StructuresProcess of Reconstruction SynthesisConclusions
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IntroductionDisplay the motions of heavenly bodiesDo the calculations of different calendarsDo the calculations of different calendarsPredict the time of eclipses
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Systematic Approach of R t ti S th iReconstruction Synthesis
aHistorical Archives
Design specifications
Topological characteristicsNumber synthesisa
Generalized Kinematic Chains
Topological characteristicsNumber synthesis
Specialized Chains
Design constraintsSpecialization
Reconstruction Designs
Ancient science and technology a
Creative Machine Design E&R Lab, National Cheng Kung University, TAIWAN
a
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Overall ContributionOverall Contribution Two examples of the lost planetary motions subsystem
are derived respectively with one and three design are derived respectively with one and three design concepts.
The inner displays the pointer of dateThe inner displays the pointer of date
41
35
The outer displays the pointer of the planetp y p
1
3
The outer displays the pointer of the planet
4 41
2
1
i b
2
4
b1
gear a1 axis baxis b
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Historical BackgroundgVenus indicator fixed wheel
80 teeth80 teeth
D 65 t th
L: 50 teeth
fixed wheel fixingD: ~65 teethrotates once per Earth year
Creative Machine Design E&R Lab, National Cheng Kung University, TAIWAN
Existing reconstruction designs from 2000 to date
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Historical BackgroundgThe feasible astronomical theory
Apollonius developed the epicycles model to Apollonius developed the epicycles model to account for the motions of five planets. And he successfully produced the retrograde motion of successfully produced the retrograde motion of planets. Planet
Epicycle
Earth
Deferent
Earth
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Deferent
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Kinematic AnalysisApollonius’ epicycles model
The planet rotates on an epicycle whose center p p yrotates on the deferent with the same direction.
d de
cosed
r
cos
1
e ded
d ee
rr
1 cose
e dedr
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Anomaly Function
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Feasible Topological Structures
Q2 Th b f li k d j i ??Q1: Input Identification of Lost Subsystem…??
Q2: The numbers of links and joints………….??Q3: The position of pin-in-slot joint…………..??
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Q1: Input Identificationp
Three feasible input conditions of lost planetary motions
Creative Machine Design E&R Lab, National Cheng Kung University, TAIWAN
Three feasible input conditions of lost planetary motions subsystem were supposed.
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Q2: Numbers of Links and Joints QThe lost planetary subsystem with NL members, NJR revolute pairs, NJA cam pairs and NJG gear pairs has one degree of pairs, NJA cam pairs and NJG gear pairs has one degree of freedom.
Based on the mobility analysis:2 3 4 0.........(1)JR JG JA LN N N N
Based on the mobility analysis
For a gear train with N joints:
0.........(2)JR JG JA JN N N N
For a gear train with NJ joints:
Each link in a geared kinematic chain must have one revolute pair:
1 0.........(3)JR LN N
have one revolute pair:
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Q2: Numbers of Links and Joints By solving above equations, the relations between th j i t d li k ld b d i d the joints and links could be derived as:
1N All Joint number2 3J LN N
1JGN
Four-bar mechanism Revolute Joint number
1JR LN N with five jointsRevolute Joint number
G i3JG LN N
2JGN
Five bar mechanism
Gear Joint number
1JAN
Five-bar mechanism with seven joints
Pin-in-slot Joint number
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JA
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Q3: Positions of Pin-in-slot Joint For a planetary gear train with n planet gears and one pin-in-slot joint:pin in slot joint:
The pin-in-slot joint is incident to the l d h li k
the rate of epicycles eT
planet gear and the output link.
N is even;the rate of epicyclethe rate of deferent
s e
pn dT
Feasible
The pin-in-slot joint is incident to two planet gears.
Feasible( 1)
( 1) the rate of deferentth t f i l
k k
ns p p o d
TT T T
TT
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the rate of epicycleks p eTT
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Design Specifications of Each Type
Type 1 Type 21. It is a planetary gear train.2. It is a four-bar mechanism
with five joints and one
1. It is a planetary gear train.2. It is a five-bar mechanism
with seven joints and one with five joints and one degree of freedom.
3. The joints include three
with seven joints and one degree of freedom.
3. The joints include four revolute joints, one external gear joint.
4. The types of mechanical
revolute joints, two external gear joints and one pin-in-slot joint.4. The types of mechanical
components are gears and links.
one pin in slot joint.4. The types of mechanical
components are gears and li klinks.
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Specialization Process of Type 1p ypIt must have at least a ground link (member 1, KF), an input link (member 2, KI), an output link (member 3, KO), and a
G d li k
( , I), p ( , O),transmission link (member 4, KT1).
STEP 1
1 1
3step 1 step 2Ground link
Input link
STEP 1
STEP 2
4step 4
step 3Output linkSTEP 3
STEP 41
2 3
1
2 3p
Transmission link
Pin-in-slot joint
STEP 4
STEP 5
2 3
4JA
2 3
4JAJG
JRstep 6
j
Revolute joints &Gear joints
STEP 6
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1 1JR JR
(a)
Gear joints
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Feasible Design Concept of Type 1Feasible Design Concept of Type 1
One feasible design concept for the inferior and One feasible design concept for the inferior and superior planets
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Specialization Process of Type 2
step 1step 1step 1 step 1step 1Ground link11
3
1
1
1
step 2 step 2step 2 step 2Not Feasible
step 2Input link1
3
1 3
1
3 1
3
step 3 step 3step 3 step 3Output link1
2
3
1 32
1
3
2
1
3
2
1
32
step 4 step 4step 4 step 4 step 4Not Feasible Not Feasible Not FeasibleTransmission link
1
2
45
33JA
1 32
4 5
JA
step 5step 5 step 5step 5Pin-in-slot joint1
2
45 JA1
2
45
JA
3 JJ
1 32
4 5
JA
J
1 32
4 5 JA
J
step 6 step 6step 6 step 6Not Feasible
j
Revolute joints & G j i t
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1
2
45 JA
JR
JR
JR
JRJG
JG
1 32
4 5
JA
JR
JR JR
JRJG
JG
1 32
4 5 JA
JR
JR JR
JR JG
JG
(a) (b) (c)
Not FeasibleGear joints
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Feasible Design Concepts of Type 2g p yp
The outer displays the pointer of the planetThe inner displays the pointer of date
41
3 5
41
2
s1
s2
b1
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axis b
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ConclusionsThis work synthesizes all feasible reconstruction
designs of the planetary subsystem through a designs of the planetary subsystem through a systematic design procedure.
aHistorical Archives
a
Design specifications
Topological characteristicsNumber synthesisa
Generalized Kinematic Chains
Design constraintsSpecialization
Specialized Chains
Ancient science and technology a
Creative Machine Design E&R Lab, National Cheng Kung University, TAIWAN
Reconstruction Designsa
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ConclusionsThe kinematic analysis of epicycles model and
the discussions of feasible topological structures the discussions of feasible topological structures were presented.
Lost subsystemLost subsystem
Input linkInput link
s1
s2
a1b1
b2
a1
b2
b1Supposed compound
gear(a) Condition 1 (b) Condition 2
Lost subsystem
Input link
a1
b2
b1
(c) Condition 3
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ContributionContribution Two examples of the lost planetary motions subsystem
are derived respectively with one and three design are derived respectively with one and three design concepts.
The inner displays the pointer of dateThe inner displays the pointer of date
41
35
The outer displays the pointer of the planetp y p
1
3
The outer displays the pointer of the planet
4 41
2
1
i b
2
4
b1
gear a1 axis baxis b
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Acknowledgementg
The authors are grateful to the National Science Council (TAIWAN ROC) under Science Council (TAIWAN, ROC) under Grant NSC 99-2221-E-006-253 for the
fi i l t f thi kfinancial support of this work.
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